New Approach to the Generalized Poincare Conjecture

Using our proof of the Poincare conjecture in dimension three and the method of mathematical induction a short and transparent proof of the generalized Poincare conjecture (the main theorem below) has been obtained. Main Theorem. Let Mn be a n-dimensional, connected, simply connected, compact, closed, smooth manifold and there exists a smooth finite triangulation on Mn which is coordinated with the smoothness structure of Mn. If Sn is the n-dimensional sphere then the manifolds Mn and Sn are homemorphic

Deformations of structures, embedding of a riemannian manifold in a kaёhlerian one and geometric antigravitation

Tubular neighborhoods play an important role in modern differential topology. The main aim of the paper is to apply these constructions to geometry of structures on Rie-mannian manifolds. Deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold are considered in section 1. In section 2, this approach is used to obtain a KaЁhlerian structure on the corresponding normal tubular neighborhood of the null section in the tangent bundle TM of a smooth manifold M. In section 3, we consider a new deformation of a tensor structure on some neighborhood ofa curve and introduce the so-called geometric antigravitation. Some results of the paper were announced in [4], [5]. The work [3] is close to our discussion

Riemannian manifolds with geometric structures

The theory of structures on manifolds is a very interesting topic of modern differential geometry and its applications. There are many results concerning various differential geometric structures on Riemannian manifolds. The main aim of this book is to get a way of a union of such results in one scheme. It seems that introduced by the author a notion of the canonical connection ⵼ and the second fundamental tensor field h adjoint to a structure is very useful for this purpose and, in many cases, it is more effective than the Riemannian connection ⵼. Especially, we pay attention to use of h to obtain classifications of structures and to the case of so-called quasi homogeneous structures. Projections of structures on submanifolds are also considered in the book

Crystal Spheres as the World

A geometric concept of the world (W) is considered where the manifold W is identified with a locally trivial fibre bundle pr: W ! U of so–called crystal spheres over a manifold U called the universal time. For every point p 2 U, Mn = pr⵼1 (p) is a n–dimensional crystal sphere and close crystal spheres are called the parallel universes. There exists a geometric black hole on the smooth manifold M n. Tensor fields, fibre bundles, operators (physical structures and equations) can be deformed towards the black hole into continuous and sectionally smooth those, further, they can be retracted together with the black hole into a small black ball to initiate the Big Bang

Embeddings of Almost Hermitian Manifold in Almost Hyper Hermitian Manifold and Complex (Hypercomplex) Numbers in Riemannian Geometry

Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold. Further, an almost hyper Hermitian structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the tangent bundle TM equipped with the deformed almost hyper Hermitian structure of the special form. As a result, we have obtained that any Riemannian manifold M of dimension n can be embedded as a totally geodesic submanifold in a Kaehlerian manifold of dimension 2n (Theorem 6) and in a hyper Kaehlerian manifold of dimension 4n (Theorem 7). Such embeddings are “good” from the point of view of Riemannian geometry. They allow solving problems of Riemannian geometry by methods of Kaehlerian geometry (see Section 5 as an example). We can find similar situation in mathematical analysis (real and complex)

A local construction of Riemannian metric

A local construction of a Riemannian metric on a smooth manifold is given by the following theorem

On almost hyperHermitian structures on Riemannian manifolds and tangent bundles

Some results concerning almost hyperHermitian structures are considered, using the notions of the canonical connection and the second fundamental tensor field h of a structure on a Riemannian manifold which were introduced by the second author. With the help of any metric connection nocebnacerutcurt s na itimreHr epyhtsomla∇ ⵼ on an almost Hermitian manifold M anstructed in the defined way on the tangent bundle TM. A similar construction was considered in [6], [7]. This structure includes two basic anticommutative almost Hermitian structures for which the second fundamental tensor fields h1 and h2 are computed. It allows us to consider various classes of almost hyperHermitian structures on TM. In particular, there exists an infinite-dimensional set of almost hyperHermitian structures on TTM where M is any Riemannian manifold

A local construction of Riemannian metric

A local construction of a Riemannian metric on a smooth manifold is given by the following theorem